Because of the translational symmetry, Lieb's theorem indicates that the fluxes \(D_i\)  must be uniform [1]}. Since \(D_i\)  commutes with the Hamiltonian \(\tilde{H}_{1D}\) , it is a constant of motion. Note that \(\eta _{\circ }^2 = \eta _{\bullet }^2=1\) , so \(D_i=\pm 1\) .
